Subgroups — C₇×#20780 #13197996

Order 1792.

758 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×#117	C₂	1	1	yes	maximal, characteristic
2	C₇×#1840	C₂	1	2	yes	maximal, complemented
3	C₇×#1831	C₂	1	2	yes	maximal, complemented
4	C₇×#1817	C₂	1	2	yes	maximal
5	C₇×#1798	C₂	1	2	yes	maximal
6	C₇×#1768	C₂	1	2	yes	maximal, complemented
7	C₇×#1750	C₂	1	2	yes	maximal
8	C₇×#1745	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1744	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₇×#117	C₂²	1	1	yes	characteristic
11	C₂×C₇×#117	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
13	C₂×C₇×C₄⋊C₈	C₂²	1	4	yes	—
14	C₂×C₇×C₂²⋊C₈	C₂²	1	4	yes	—
15	C₂×C₇×#97	C₂²	1	2	yes	—
16	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
17	C₄×C₇×C₄⋊C₄	C₄	1	2	yes	—
18	C₄×C₇×C₄⋊C₄	C₄	1	2	yes	—
19	C₄×C₇×C₂²⋊C₄	C₄	1	2	yes	—
20	C₄×C₇×C₂²⋊C₄	C₄	1	2	yes	—
21	C₇×#376	C₂²	1	1	yes	characteristic
22	C₇×#376	C₂²	1	1	yes	characteristic
23	C₄³×C₇	C₂²	1	1	yes	characteristic
24	C₄³×C₇	C₂²	1	1	yes	characteristic
25	C₇×#336	C₂²	1	2	yes	—
26	C₇×#336	C₂²	1	2	yes	—
27	C₇×#336	C₂²	1	2	yes	—
28	C₇×#336	C₂²	1	2	yes	—
29	C₇×#336	C₂²	1	4	yes	—
30	C₇×#336	C₂²	1	4	yes	—
31	#20780	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
34	C₂²×C₇×C₈	C₂³	1	4	yes	—
35	C₂²×C₇×C₈	C₂³	1	4	yes	—
36	C₇×#117	—	2	4	no	—
37	C₇×#117	—	2	4	no	—
38	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
39	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
40	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
41	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
42	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
43	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₇	—	2	4	no	—
45	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
56	C₇×C₄⋊C₈	—	2	8	no	—
57	C₇×C₂²⋊C₈	—	2	8	no	—
58	C₇×#97	—	2	4	no	—
59	C₄×C₇×C₈	—	2	4	no	—
60	C₇×#95	C₂×C₄	1	2	yes	—
61	C₇×#95	C₂×C₄	1	2	yes	—
62	C₇×#95	C₂×C₄	1	2	yes	—
63	C₇×#95	C₂×C₄	1	2	yes	—
64	C₄×#117	C₂×C₇	1	1	yes	characteristic
65	#1840	C₂×C₇	1	2	yes	complemented
66	#1831	C₂×C₇	1	2	yes	complemented
67	#1817	C₂×C₇	1	2	yes	—
68	#1798	C₂×C₇	1	2	yes	—
69	#1768	C₂×C₇	1	2	yes	complemented
70	#1750	C₂×C₇	1	2	yes	—
71	#1745	C₂×C₇	1	1	yes	characteristic, complemented
72	#1744	C₂×C₇	1	1	yes	characteristic, complemented
73	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₇	—	2	2	no	—
75	C₂²×C₄×C₇	—	2	2	no	—
76	C₂²×C₄×C₇	—	2	4	no	—
77	C₂²×C₄×C₇	—	2	4	no	—
78	C₂²×C₄×C₇	—	4	2	no	—
79	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₇	#41	1	4	yes	—
89	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
90	C₂×C₇×C₈	—	2	4	no	—
91	C₂×C₇×C₈	—	2	4	no	—
92	C₂×C₇×C₈	—	2	4	no	—
93	C₂×C₇×C₈	—	2	4	no	—
94	C₂×C₇×C₈	—	2	4	no	—
95	C₂×C₇×C₈	—	2	4	no	—
96	C₇×C₄⋊C₄	—	2	2	no	—
97	C₇×C₄⋊C₄	—	2	2	no	—
98	C₇×C₄⋊C₄	—	2	4	no	—
99	C₇×C₂²⋊C₄	—	2	2	no	—
100	C₇×C₂²⋊C₄	—	2	2	no	—
101	C₇×C₂²⋊C₄	—	2	4	no	—
102	C₄²×C₇	—	2	4	no	—
103	C₄²×C₇	—	2	4	no	—
104	C₄²×C₇	—	2	4	no	—
105	C₄²×C₇	—	2	2	no	—
106	C₄²×C₇	—	2	2	no	—
107	C₄²×C₇	—	2	2	no	—
108	C₄²×C₇	—	2	2	no	—
109	C₄²×C₇	—	4	2	no	—
110	C₄²×C₇	—	4	2	no	—
111	C₄²×C₇	#41	1	4	yes	—
112	C₄²×C₇	#41	1	4	yes	—
113	C₂×#117	C₂²×C₇	1	1	yes	characteristic
114	C₂×#117	C₂²×C₇	1	1	yes	characteristic
115	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
116	C₂×C₄⋊C₈	C₂²×C₇	1	4	yes	—
117	C₂×C₂²⋊C₈	C₂²×C₇	1	4	yes	—
118	C₂×#97	C₂²×C₇	1	2	yes	—
119	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
120	C₄×C₄⋊C₄	C₄×C₇	1	2	yes	—
121	C₄×C₄⋊C₄	C₄×C₇	1	2	yes	—
122	C₄×C₂²⋊C₄	C₄×C₇	1	2	yes	—
123	C₄×C₂²⋊C₄	C₄×C₇	1	2	yes	—
124	#376	C₂²×C₇	1	1	yes	characteristic
125	#376	C₂²×C₇	1	1	yes	characteristic
126	C₄³	C₂²×C₇	1	1	yes	characteristic
127	C₄³	C₂²×C₇	1	1	yes	characteristic
128	#336	C₂²×C₇	1	2	yes	—
129	#336	C₂²×C₇	1	2	yes	—
130	#336	C₂²×C₇	1	2	yes	—
131	#336	C₂²×C₇	1	2	yes	—
132	#336	C₂²×C₇	1	4	yes	—
133	#336	C₂²×C₇	1	4	yes	—
134	C₂³×C₇	—	2	1	no	—
135	C₂³×C₇	—	2	1	no	—
136	C₂³×C₇	—	2	2	no	—
137	C₂³×C₇	—	2	2	no	—
138	C₂³×C₇	#117	1	2	yes	—
139	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₇	—	2	2	no	—
141	C₂×C₄×C₇	—	2	1	no	—
142	C₂×C₄×C₇	—	2	1	no	—
143	C₂×C₄×C₇	—	2	2	no	—
144	C₂×C₄×C₇	—	2	2	no	—
145	C₂×C₄×C₇	—	2	2	no	—
146	C₂×C₄×C₇	—	2	2	no	—
147	C₂×C₄×C₇	—	2	2	no	—
148	C₂×C₄×C₇	—	2	8	no	—
149	C₂×C₄×C₇	—	2	2	no	—
150	C₂×C₄×C₇	—	2	1	no	—
151	C₂×C₄×C₇	—	2	1	no	—
152	C₂×C₄×C₇	—	2	2	no	—
153	C₂×C₄×C₇	—	2	2	no	—
154	C₂×C₄×C₇	—	2	2	no	—
155	C₂×C₄×C₇	—	2	2	no	—
156	C₂×C₄×C₇	—	2	2	no	—
157	C₂×C₄×C₇	—	2	2	no	—
158	C₂×C₄×C₇	—	2	2	no	—
159	C₂×C₄×C₇	—	4	8	no	—
160	C₂×C₄×C₇	#117	1	2	yes	—
161	C₂×C₄×C₇	#117	1	2	yes	—
162	C₂×C₄×C₇	#117	1	2	yes	—
163	C₂×C₄×C₇	#117	1	4	yes	—
164	C₂×C₄×C₇	#131	1	4	yes	—
165	C₂×C₄×C₇	C₂×#41	1	2	yes	—
166	C₂×C₄×C₇	C₂×#41	1	2	yes	—
167	C₂×C₄×C₇	C₂×#41	1	2	yes	—
168	C₇×C₈	—	4	4	no	—
169	C₇×C₈	—	4	4	no	—
170	C₂³×C₄	C₂×C₄×C₇	1	2	yes	—
171	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
172	C₂²×C₈	C₂³×C₇	1	4	yes	—
173	C₂²×C₈	C₂³×C₇	1	4	yes	—
174	#117	—	2	4	no	—
175	#117	—	2	4	no	—
176	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
177	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
178	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
179	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
180	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
181	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
182	C₂×C₄²	—	2	4	no	—
183	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
184	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
185	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
186	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
187	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
188	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
189	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
190	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
191	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
192	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
193	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
194	C₄⋊C₈	—	2	8	no	—
195	C₂²⋊C₈	—	2	8	no	—
196	#97	—	2	4	no	—
197	C₄×C₈	—	2	4	no	—
198	#95	C₂×C₄×C₇	1	2	yes	—
199	#95	C₂×C₄×C₇	1	2	yes	—
200	#95	C₂×C₄×C₇	1	2	yes	—
201	#95	C₂×C₄×C₇	1	2	yes	—
202	C₂²×C₇	—	2	4	no	—
203	C₂²×C₇	—	2	2	no	—
204	C₂²×C₇	—	4	1	no	—
205	C₂²×C₇	—	4	1	no	—
206	C₂²×C₇	—	4	2	no	—
207	C₂²×C₇	C₂×#117	1	1	yes	characteristic
208	C₂²×C₇	C₂×#117	1	1	yes	characteristic
209	C₂²×C₇	C₂×#117	1	1	yes	characteristic
210	C₂²×C₇	C₂×#131	1	2	yes	—
211	C₂²×C₇	#568	1	2	yes	—
212	C₄×C₇	—	2	2	no	—
213	C₄×C₇	—	2	2	no	—
214	C₄×C₇	—	2	4	no	—
215	C₄×C₇	—	2	4	no	—
216	C₄×C₇	—	2	2	no	—
217	C₄×C₇	—	4	1	no	—
218	C₄×C₇	—	4	1	no	—
219	C₄×C₇	—	4	2	no	—
220	C₄×C₇	—	4	2	no	—
221	C₄×C₇	#529	1	4	yes	—
222	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	4	2	no	—
228	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
229	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
230	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
231	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
232	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
233	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
234	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
235	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
236	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
237	C₂²×C₄	C₇×#41	1	4	yes	—
238	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	4	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₂²⋊C₄	—	2	2	no	—
249	C₂²⋊C₄	—	2	2	no	—
250	C₂²⋊C₄	—	2	4	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	2	4	no	—
253	C₄²	—	2	4	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	4	2	no	—
259	C₄²	—	4	2	no	—
260	C₄²	C₇×#41	1	4	yes	—
261	C₄²	C₇×#41	1	4	yes	—
262	C₂×C₇	—	4	2	no	—
263	C₂×C₇	#2283	1	1	yes	characteristic
264	C₂×C₇	#2922	1	1	yes	characteristic
265	C₂×C₇	#2923	1	1	yes	characteristic
266	C₂×C₇	#2924	1	2	yes	—
267	C₂×C₇	#2925	1	2	yes	—
268	C₂³	—	2	1	no	—
269	C₂³	—	2	1	no	—
270	C₂³	—	2	2	no	—
271	C₂³	—	2	2	no	—
272	C₂³	C₇×#117	1	2	yes	—
273	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	8	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	8	no	—
294	C₂×C₄	C₇×#117	1	2	yes	—
295	C₂×C₄	C₇×#117	1	2	yes	—
296	C₂×C₄	C₇×#117	1	2	yes	—
297	C₂×C₄	C₇×#117	1	4	yes	—
298	C₂×C₄	C₇×#131	1	4	yes	—
299	C₂×C₄	C₂×C₇×#41	1	2	yes	—
300	C₂×C₄	C₂×C₇×#41	1	2	yes	—
301	C₂×C₄	C₂×C₇×#41	1	2	yes	—
302	C₈	—	4	4	no	—
303	C₈	—	4	4	no	—
304	C₇	#20780	1	1	yes	characteristic, direct product factor, Sylow
305	C₂²	—	2	4	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	4	2	no	—
310	C₂²	C₂×C₇×#117	1	1	yes	characteristic
311	C₂²	C₂×C₇×#117	1	1	yes	characteristic
312	C₂²	C₂×C₇×#117	1	1	yes	characteristic
313	C₂²	C₂×C₇×#131	1	2	yes	—
314	C₂²	C₇×#568	1	2	yes	—
315	C₄	—	2	2	no	—
316	C₄	—	2	2	no	—
317	C₄	—	2	4	no	—
318	C₄	—	2	4	no	—
319	C₄	—	2	2	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	2	no	—
323	C₄	—	4	2	no	—
324	C₄	C₇×#529	1	4	yes	—
325	C₂	—	4	2	no	—
326	C₂	C₇×#2283	1	1	yes	characteristic
327	C₂	C₇×#2922	1	1	yes	characteristic
328	C₂	C₇×#2923	1	1	yes	characteristic
329	C₂	C₇×#2924	1	2	yes	—
330	C₂	C₇×#2925	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂³×C₇	C₂³×C₄
prime radical	C₂⁴×C₇	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₇×#1840	C₂	semidirect
3. C₇×#1831	C₂	semidirect
6. C₇×#1768	C₂	semidirect
8. C₇×#1745	C₂	semidirect
9. C₇×#1744	C₂	semidirect
31. #20780	C₇	direct (N × H)
65. #1840	C₂×C₇	semidirect
66. #1831	C₂×C₇	semidirect
69. #1768	C₂×C₇	semidirect
71. #1745	C₂×C₇	semidirect
72. #1744	C₂×C₇	semidirect
304. C₇	#20780	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×C₇×#117	2
C₇×#1840	2
C₇×#1831	2
C₇×#1817	2
C₇×#1798	2
C₇×#1768	2
C₇×#1750	2
C₇×#1745	2
C₇×#1744	2
#20780	7

Subgroups — C7×#20780 #13197996

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#20780 #13197996